Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

Bugeaud, Yann; Mignotte, Maurice; Siksek, Samir

Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers

Authors

Bugeaud, Yann

Mignotte, Maurice

Siksek, Samir

Description

This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.